In Section 4, we consider the restriction operator \(\mathcal {R}f = \widehat{f} \, |_{\Sigma }\) , with \(\Sigma \)  given by
(REF ) (with \(U = V^{c,d}_{1}\) ). Following ideas in [1]}, using (REF ) and complex interpolation for a suitable analytic family of operators, in Theorem REF , we prove that there exists a constant \(C >0\)  such that
\(\Vert  \mathcal {R}f \Vert _{L^{2}(\Sigma )} \le C \, \Vert  f \Vert _{L^{p}(\mathbb {R}^{4})}\)  for all \(f \in \mathcal {S}(\mathbb {R}^{4})\)  if and only if
\(\displaystyle {\frac{\alpha _1 + \alpha _2 + 4m}{2(\alpha _1 + \alpha _2 + 2m)}} \le \frac{1}{p} \le 1\) . Finally, using again (REF ),
we study the type set \(E_{\mu }\)  for the convolution operator \(T_{\mu }f = \mu \ast f\) , for \(f \in \mathcal {S}(\mathbb {R}^{4})\) . Theorem
REF  states that, for \(p = \displaystyle {\frac{\alpha _1 + \alpha _2 + 2m}{\alpha _1 + \alpha _2 + m}}\) , the closed triangle with vertices \((0,0)\) , \((1,1)\)  and \(\left(\frac{1}{p}, \frac{1}{p^{\prime }} \right)\)  is contained in \(E_{\mu }\) ; moreover
\(\left(\frac{1}{p}, \frac{1}{p^{\prime }} \right) \in \partial E_{\mu }\) . Finally, in Section 5, we give an example of a function
\(\varphi =(\varphi _1, \varphi _2) : V^{a,b} \rightarrow \mathbb {R}^{2}\)  satisfying the hypotheses H1 - H4.
